# Metamath Proof Explorer

## Theorem minmar1fval

Description: First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018)

Ref Expression
Hypotheses minmar1fval.a 𝐴 = ( 𝑁 Mat 𝑅 )
minmar1fval.b 𝐵 = ( Base ‘ 𝐴 )
minmar1fval.q 𝑄 = ( 𝑁 minMatR1 𝑅 )
minmar1fval.o 1 = ( 1r𝑅 )
minmar1fval.z 0 = ( 0g𝑅 )
Assertion minmar1fval 𝑄 = ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )

### Proof

Step Hyp Ref Expression
1 minmar1fval.a 𝐴 = ( 𝑁 Mat 𝑅 )
2 minmar1fval.b 𝐵 = ( Base ‘ 𝐴 )
3 minmar1fval.q 𝑄 = ( 𝑁 minMatR1 𝑅 )
4 minmar1fval.o 1 = ( 1r𝑅 )
5 minmar1fval.z 0 = ( 0g𝑅 )
6 oveq12 ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = ( 𝑁 Mat 𝑅 ) )
7 6 1 eqtr4di ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = 𝐴 )
8 7 fveq2d ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = ( Base ‘ 𝐴 ) )
9 8 2 eqtr4di ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = 𝐵 )
10 simpl ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → 𝑛 = 𝑁 )
11 fveq2 ( 𝑟 = 𝑅 → ( 1r𝑟 ) = ( 1r𝑅 ) )
12 11 4 eqtr4di ( 𝑟 = 𝑅 → ( 1r𝑟 ) = 1 )
13 fveq2 ( 𝑟 = 𝑅 → ( 0g𝑟 ) = ( 0g𝑅 ) )
14 13 5 eqtr4di ( 𝑟 = 𝑅 → ( 0g𝑟 ) = 0 )
15 12 14 ifeq12d ( 𝑟 = 𝑅 → if ( 𝑗 = 𝑙 , ( 1r𝑟 ) , ( 0g𝑟 ) ) = if ( 𝑗 = 𝑙 , 1 , 0 ) )
16 15 ifeq1d ( 𝑟 = 𝑅 → if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r𝑟 ) , ( 0g𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) = if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) )
17 16 adantl ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r𝑟 ) , ( 0g𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) = if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) )
18 10 10 17 mpoeq123dv ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑖𝑛 , 𝑗𝑛 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r𝑟 ) , ( 0g𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) ) = ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) )
19 10 10 18 mpoeq123dv ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑘𝑛 , 𝑙𝑛 ↦ ( 𝑖𝑛 , 𝑗𝑛 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r𝑟 ) , ( 0g𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) ) ) = ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
20 9 19 mpteq12dv ( ( 𝑛 = 𝑁𝑟 = 𝑅 ) → ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) ↦ ( 𝑘𝑛 , 𝑙𝑛 ↦ ( 𝑖𝑛 , 𝑗𝑛 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r𝑟 ) , ( 0g𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
21 df-minmar1 minMatR1 = ( 𝑛 ∈ V , 𝑟 ∈ V ↦ ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) ↦ ( 𝑘𝑛 , 𝑙𝑛 ↦ ( 𝑖𝑛 , 𝑗𝑛 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , ( 1r𝑟 ) , ( 0g𝑟 ) ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
22 2 fvexi 𝐵 ∈ V
23 22 mptex ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V
24 20 21 23 ovmpoa ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 minMatR1 𝑅 ) = ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
25 21 mpondm0 ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 minMatR1 𝑅 ) = ∅ )
26 mpt0 ( 𝑚 ∈ ∅ ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ∅
27 25 26 eqtr4di ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 minMatR1 𝑅 ) = ( 𝑚 ∈ ∅ ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
28 1 fveq2i ( Base ‘ 𝐴 ) = ( Base ‘ ( 𝑁 Mat 𝑅 ) )
29 2 28 eqtri 𝐵 = ( Base ‘ ( 𝑁 Mat 𝑅 ) )
30 matbas0pc ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( Base ‘ ( 𝑁 Mat 𝑅 ) ) = ∅ )
31 29 30 syl5eq ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ )
32 31 mpteq1d ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ( 𝑚 ∈ ∅ ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
33 27 32 eqtr4d ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 minMatR1 𝑅 ) = ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) )
34 24 33 pm2.61i ( 𝑁 minMatR1 𝑅 ) = ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )
35 3 34 eqtri 𝑄 = ( 𝑚𝐵 ↦ ( 𝑘𝑁 , 𝑙𝑁 ↦ ( 𝑖𝑁 , 𝑗𝑁 ↦ if ( 𝑖 = 𝑘 , if ( 𝑗 = 𝑙 , 1 , 0 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) )